Black-Box Identity Testing of Noncommutative Rational Formulas in Deterministic Quasipolynomial Time

TL;DR

This paper presents the first quasipolynomial-size hitting set for all polynomial-size noncommutative rational formulas, advancing deterministic black-box identity testing and white-box complexity understanding.

Contribution

It introduces the first quasipolynomial-size hitting set for all rational formulas of polynomial size, improving black-box RIT complexity and providing a deterministic quasi-NC upper bound.

Findings

First quasipolynomial-size hitting set for all rational formulas

Deterministic quasi-NC upper bound for white-box RIT

Significant progress on black-box RIT complexity

Abstract

Rational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg, Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018); Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen and Makam (2017)] in the black-box setting, via singularity testing of linear matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in the white-box setting follows from the result of Derksen and Makam (2017). Designing an efficient deterministic black-box algorithm for RIT and understanding the parallel complexity of RIT are major open problems in this area. Despite being open since the work of Garg, Gurvits, Oliveira, and Wigderson (2016), these questions have seen limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [Arvind, Chatterjee, Mukhopadhyay (2022)]. In this paper, we significantly improve the black-box complexity of this problem and obtain the first quasipolynomial-size hitting set for all rational formulas of polynomial size. Our construction also yields the first deterministic quasi-NC upper bound for RIT in the white-box setting.